Deep-Sea Research II 46 (1999) 1385}1425
On the parameterisation of oceanic sensible heat
loss to the atmosphere and to ice in an ice-covered
mixed layer in winter
Bert Rudels *, Hans J. Friedrich, Dagmar Hainbucher,
Gerrit Lohmann
Finnish Institute of Marine Research, PL33, FIN-00931 Helsinki, Finland
Institut fu( r Meereskunde der Universita( t Hamburg, Troplowitzstra}e 7, D-22529 Hamburg, Germany
Alfred Wegener Institut fu( r Polar- und Meeresforschung, Postfach 120161, D-27515, Bremerhaven, Germany
Received 11 February 1998; received in revised form 3 November 1998; accepted 5 November 1998
Abstract
In high-latitude oceans with seasonal ice cover, the ice and the low-salinity mixed layer form
an interacting barrier for the heat #ux from the ocean to the atmosphere. The presence of a less
dense surface layer allows ice to form, and the ice cover reduces the heat loss to the atmosphere.
The ice formation weakens the stability at the base of the mixed layer, leading to stronger
entrainment and larger heat #ux from below. This heat transport retards, and perhaps stops, the
growth of the ice cover. As much heat is then entrained from below as is lost to the atmosphere.
This heat loss further reduces the stability, and unless a net ice melt occurs, the mixed layer
convects. Two possibilities exist: (1) A net ice melt, su$cient to retain the stability, will always
occur and convection will not take place until all ice is removed. The deep convection will then
be thermal, deepening the mixed layer. (2) The ice remains until the stability at the base of the
mixed layer disappears. The mixed layer then convects, through haline convection, into the
deep ocean. Warm water rises towards the surface and the ice starts to melt, and a new mixed
layer is reformed. The present work discusses the interactions between ice cover and entrainment during winter, when heat loss to the atmosphere is present. One crucial hypothesis is
introduced: `When ice is present and the ocean loses sensible heat to the atmosphere and to ice
melt, the buoyancy input at the sea surface due to ice melt is at a minimuma. Using a onedimensional energy-balance model, applied to the arti"cial situation, where ice melts directly on
warmer water, it is found that this corresponds to a constant fraction of the heat loss going to
ice melt. It is postulated that this partitioning holds for the ice cover and the mixed layer in the
high-latitude ocean. When a constant fraction of heat goes to ice melt, at least one deep
* Corresponding author. Fax: 00358-9-331-025.
E-mail address: rudels@"mr." (B. Rudels)
0967-0645/99/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.
PII: S 0 9 6 7 - 0 6 4 5 ( 9 9 ) 0 0 0 2 8 - 4
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B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
convection event occurs, before the ice cover can be removed by heat entrained from below.
After one or several convection events the ice normally disappears and a deep-reaching thermal
convection is established. Conditions appropriate for the Weddell Sea and the Greenland Sea
are examined and compared with "eld observations. With realistic initial conditions no
convection occurs in the warm regime of the Weddell Sea. A balance between entrained heat
and atmospheric heat loss is established and the ice cover remains throughout the winter. At
Maud Rise convection may occur, but late in winter and normally no polynya can form before
the summer ice melt. In the central Greenland Sea the mixed layer generally convects early in
winter and the ice is removed by melting from below as early as February or March. This is in
agreement with existing observations. 1999 Elsevier Science Ltd. All rights reserved.
1. Introduction
High-latitude oceans in both hemispheres are covered with sea ice. Part of the ice
cover is perennial, surviving the summer, but most of it is seasonal, forming in fall and
disappearing the following summer. The fraction of permanent ice cover is largest in
the Arctic, where the ratio of minimum to maximum ice cover extent is about 1/2. In
the southern ocean only 1/5 of the winter ice cover survives the following summer.
This di!erence is mainly due to the con"ned geography of the Arctic Ocean, which
restricts the export of ice to lower latitudes, where the ice diverges and where
a stronger input of sensible heat from the underlying ocean is present. If the ice drifts
into warmer water, this heat could come directly from the ocean. It also could be due
to a larger presence of leads in a diverging ice cover, allowing the solar radiation to
become absorbed in the water column. A second factor, acting to preserve the ice
cover in the Arctic Ocean, is the strong strati"cation of the water column due to the
low-salinity Polar Mixed Layer and the intermediate, cold halocline. The presence of
the halocline over a large part of the Arctic Ocean leads to entrainment of colder
water. Should the mixed layer deepen by freezing and haline convection during winter,
no heat is added, reducing the ice formation. The limiting factor for the ice growth is
the ice thickness, which diminishes the heat loss to the atmosphere.
In the Greenland Sea, as also in the Weddell Sea, ice has been observed to disappear
in late winter, when heat loss to the atmosphere still occurs and no input of short wave
radiation is present. In the Greenland Sea this has been partly attributed to winds
removing the ice. However, in some instances the removal has been so rapid that
melting, caused by sensible heat supplied from below, is a more probable explanation.
This is also likely for the Weddell Sea polynya, which was observed in late winter in
the mid-1970s (Zwally et al., 1983). The observation of a deep ('4000 m),
homogenous water column in the following summer suggested that the polynya had
either been preceded by, or been the cause of, deep reaching convection (Gordon,
1978).
The water columns in the central Greenland and Weddell Seas are weakly strati"ed,
and a cold, less saline surface layer rides on top of warmer, more saline water.
A reservoir of sensible heat is close and may be tapped in winter. That this heat is not
immediately available is clear from the fact that the cooling in autumn leads to
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1387
freezing temperature and ice formation. Not until late in winter will the heat #ux from
the underlying water become large enough to start to melt the ice. The strati"cation in
fall thus prevents the heat #ux from below from balancing the heat lost to the
atmosphere, and ice must form. Freezing leads to brine rejection. The density of the
surface layer increases, and the stability of the water column decreases. The growth of
the ice cover reduces the heat loss to the atmosphere, and if the initial strati"cation of
the water column is su$ciently weak, the heat #ux from below may increase su$ciently to balance the heat lost to the atmosphere and stop further ice growth. The
subsequent disappearance of the ice in weakly strati"ed areas, and the deep almost
homogenous water columns observed in spring suggest that convection has taken
place. A convective removal of the upper layer would bring warm water into contact
with the ice cover and cause the ice to melt, and perhaps to disappear. Such
convection, since the mixed layer is at freezing temperature, has to be haline.
However, Backhaus (1995), applying a non-hydrostatic convection model for haline
convection in the Greenland Sea, found that as the stability decreases, plumes
reaching the lower boundary of the mixed layer would induce such strong entrainment of warmer water into the mixed layer that it would practically inhibit the ice
formation and the generation of saline, dense plumes. The convection would become
weaker and the deepening of the mixed layer would stop. Backhaus then suggested
that the ice has to be largely removed by wind before a deeper-reaching, thermal
convection could be established. Observations of convection occurring in regions
where no mechanically induced shear is present have normally indicated less energetic
entrainment. The mixed layer becomes homogenised by the convection, but the
entrainment due to convection is signi"cantly smaller than that caused by windgenerated turbulence (Farmer, 1975). The strong vertical heat #ux obtained by
Backhaus (1995) could be due to the use of a non-hydrostatic convection model, which
leads to larger vertical displacements as the sinking plumes impinge upon the stable
interface at the bottom of the mixed layer.
In most mixed-layer models it is assumed that turbulent entrainment caused by
wind is signi"cantly larger than that due to convection, and Walin (1993) studied the
e!ects of wind-induced turbulence on the ice cover and the entrainment. Adopting
a simple energy balance model (Niiler and Kraus, 1977) he assumed that the development of the high-latitude mixed layer in winter could be considered in two idealised
stages. In the "rst stage there is little (no) entrainment, and the salinity (and density)
of the mixed layer increases due to freezing and brine rejection. In the second phase
the stability has been lowered su$ciently for a `freeze-meltinga state to be attained,
where heat entrained from below supplies all the heat lost to the atmosphere. Heat
is also available to melt enough ice to keep the upper layer stable as it deepens. Given
a long enough winter the ice would eventually disappear. The stabilising freshwater
input is then gone and the upper layer convects, thermally, to great depths. In
the Weddell Sea, which was the focus of Walin's discussion, the duration of the freezemelting period has to be signi"cantly longer than the span of the winter to remove
the ice, and the ice cover would normally remain and melt the following summer.
Visbeck et al. (1995) applied a similar model to the Greenland Sea. They derived
a similar pattern of ice growth and melting and, assuming a partial export of the ice
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B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
cover, a situation arose where the ice disappeared and the mixed layer deepened
rapidly.
The results from these studies then imply: (1) Deep haline convection is not possible,
because heat entrained from below will inhibit the ice formation; (2) ice is removed by
melting and/or exported before deep convection occurs; (3) all deep convection in the
ocean is thermal.
For one of us (BR), being a supporter of the idea that deep haline convection is the
most important process for triggering the deep convection events forming the Greenland Sea Deep Water (Rudels, 1990,1993), and perhaps also the deep, vertically
homogenous water column associated with the Weddell Sea polynya, these are
disturbing results. Before any fruitless e!orts on describing haline deep convection are
made, the e!ects of the heat #ux from the underlying ocean on the evolution of the
mixed layer, on the entrainment, and on the ice cover, must be considered. Following
conventional wisdom, we assume that wind stirring is more e$cient in entraining
water into the mixed layer and thereby concentrate on the e!ects of mechanically
generated entrainment on the ice growth and do not study the strongly convective
situation considered by Backhaus (1995).
The formulation of the buoyancy #ux used by Walin (1993) and Visbeck et al. (1995)
is di!erent from the one applied by Lemke (1987), Lemke et al. (1990), Houssais (1988)
and Houssais and Hibler (1993). Walin (1993) balances entrained heat with heat lost
to the atmosphere. If it is smaller, ice is formed. If it is greater, ice melts. In the "rst
phase all entrained heat goes directly to the atmosphere; in the second, freeze-melting
phase, no new ice is formed although the mixed layer is at freezing temperature. The
entrainment is su$ciently strong to provide, directly, all heat to the atmosphere and
some additional heat to melt ice. This freshwater input is necessary to maintain the
strati"cation and to prevent the mixed layer from overturning.
By contrast, Lemke (1987) allows all heat entrained into the mixed layer to go to
melting. The heat lost to the atmosphere is balanced by ice formation, in leads and
under the ice #oes. The buoyancy #ux then has two contributions, one positive due to
ice melt and related to the entrainment from below, and one negative from ice
formation, determined by the heat loss to the atmosphere. This separation does not
change the ice balance or the stability at the base of the mixed layer. However, if
entrainment caused by convection is weaker than that due to wind generated turbulence, the reduced entrainment, which results from increased melting, cannot be
balanced by entrainment driven by haline convection. The heat #ux into the mixed
layer from below will be smaller and more ice must form to balance the heat loss to the
atmosphere.
Lemke (1987) and Houssais (1988) do not explicitly address the e!ects of entrainment upon the ice cover and of ice melt upon the entrainment. Lemke (1987)
considered primarily the situation in the Arctic Ocean, but in a later work (Lemke
et al., 1990) exponential pro"les for vertical temperature and salinity distributions
beneath the mixed layer were used from the Weddell Sea. This means that deepening
but no overturning occurs, and the issue of net melting providing the buoyancy input
needed to keep the strati"cation, and the mixed layer, does not arise. Houssais (1988)
and Houssais and Hibler (1993) studied the Greenland Sea, and Houssais (1988) found
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1389
that the deepest convection, down to 3000 m in some cases, occurs in the ice-free area
in the central Greenland Sea. However, it is di$cult to determine from her "gures, if
the ice edge is controlled by entrainment and melting of the ice by heat from below, or
if it is related to the initial distribution of low-salinity surface water.
Which formulation is the most realistic one? For the melt water to inhibit the
entrainment e!ectively, it must be mixed away from the ice into the mixed layer.
A melting and subsequent freezing within the ice cover would not lead to a buoyancy
#ux. We therefore expect that some, but not all, entrained heat will go directly to the
atmosphere without causing any melt water to enter the mixed layer. This fraction
would be somewhere between the 0 of the Lemke (1987) formulation and the 1, which
holds for the "rst phase in the model by Walin (1993), when the heat loss to the
atmosphere is larger than the heat #ux from below. If the situation arises that the heat
#ux from below is larger than the heat loss to the atmosphere, as in the freeze-melt
stage (Walin, 1993), we still expect ice to form. When the ice cover becomes thinner
and less compact, the likelihood for new ice formation may increase, even if there is
a net ice melt. To balance the heat #uxes, more than the excess entrained heat will go
to ice melt.
We propose a rationale for determining the distribution of the loss of oceanic
sensible heat to ice melt and to the atmosphere. We assume that the entrainment can
be described by an energy balance equation similar to the one used by Walin (1993).
As with Walin (1993) the dissipation of the energy is taken to be independent of the
depth of the mixed layer, although an exponential decay, like the one used by
Houssais (1988), probably is more realistic. This would, however, bias our work
against entrainment and make comparison with, e.g. Walin (1993) more di$cult. For
the same reason, as well as for simplicity, we assume an idealised two-layer ocean with
an homogeneous mixed layer and an homogeneous deep ocean. The turbulent
di!usion through the pycnocline is considered weak compared to the entrainment and
is ignored. These assumptions also favour entrainment compared to the case with
a strati"ed deep ocean (Lemke et al., 1990).
Only the cooling situation, with a cold, windy atmosphere above the ocean,
is considered and we introduce the following hypothesis: When ice is present and
the ocean loses sensible heat both to the atmosphere and to ice melt, the heat loss
is so distributed that the buoyancy input at the sea surface due to ice melt is at
a minimum. This holds regardless of the compactness and the thickness of the ice
cover as long as it is horizontally homogeneous and the situation can be viewed as
one-dimensional.
The hypothesis implies that the smallest possible part of the mechanical energy
imparted to the mixed layer by the wind will go to stirring melt water into the mixed
layer. Consequently the largest possible amount will be available for entrainment. The
hypothesis thus maximises the #ux of oceanic sensible heat to the mixed layer.
Optimising methods have been used in several aspects of physics. We are well aware
that its use here cannot be compared to the variational principles generally applied in
mechanics. However, nature often seems to approach some optimal state. The problem is to understand what is being optimised. Be that as it may, at a more profane
level, an extremum singles out one value among an in"nity of possible ones and thus
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B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
provides one reason for choosing this value. It is, at least in one respect, di!erent from
all the rest.
To "nd the partitioning for which the ice melt is a minimum, we must examine
a situation where no latent heat of freezing enters the heat balance. We therefore
consider a thought experiment: Ice is placed on top of a vertically homogeneous water
column with temperature above freezing. Heat is lost by the ocean to the atmosphere
and to ice melt. The melt water is stirred into the water column, and an upper mixed
layer begins to form. The fraction of heat going to ice melt, which produces the
smallest buoyancy input, can then be determined (Section 2.1). Section 2.2 then
considers a situation like the one encountered in the Greenland Sea, where a less
saline surface layer, lying above warmer water, is brought to freezing temperature in
autumn and ice starts to form. The heat entrained into the upper layer from below is
assumed distributed according to the partitioning obtained in Section 2.1. If cooling,
freezing and brine release should remove the density step at the base of the mixed layer
before the ice is melted, convection will bring the water of the mixed layer into the
deep ocean. Warm water rises to the surface, ice starts to melt and a new mixed layer
forms. Note that if the ice melts before the stability is gone, as was found by Walin
(1993), this convection will not happen. Only situations with heat loss to the atmosphere are considered, and Section 3 describes the parameterisations of the heat loss
and the evolution of the mixed layer which are used, and the e!ects of di!erent forcing
and strati"cation in some idealised cases before the situations in the Weddell Sea
(Section 4.2) and the Greenland Sea (Section 4.3) are then considered.
2. Ice melt, ice formation, entrainment and convection
2.1. Ice melting on warmer water
Assume heat loss to the atmosphere, an horizontally homogeneous ice "eld, and
apply an one-dimensional model for the water column. The salt and heat #uxes at the
upper boundary of the ocean are
o
o
o
o
"! M (S!S )# F (S!S )# ES! PS,
X
G
G
o
o o
o
Q
1
w¹" "! "! +Q #o M [¸#c(¹!¹ )],.
(1)
X
o c
o c T and S are the temperature and salinity at the sea surface, M is ice melted, F is ice
formed, E and P denote evaporation and precipitation, S is the salinity of sea ice and
G
¹ the freezing temperature, o , o and o are the densities of freshwater, ice and sea
D
water, respectively, Q is the loss of sensible heat by the ocean and Q is the part going
to atmosphere and space, which includes #uxes of sensible heat and latent heat of
evaporation and net long wave radiation. We consider a winter situation and contributions from short-wave radiation are ignored. L is the latent heat of melting, and
c the speci"c heat of sea water.
wS"
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1391
To concentrate on the partitioning of the oceanic heat loss we make some simpli"cations. Precipitation and evaporation are ignored in the salt balance. The sensible
heat stored in the ice and the heat capacity of the melt water are considered small and
neglected compared with the latent heat of melting. In most situations the mean ice
temperature is about !103, and the sensible heat in the ice amounts to 5% of the
latent heat of melting. If the temperature of the ocean is 1}23C above freezing, the
heat needed to rise the temperature of the melt water by this amount corresponds to
2% of the heat needed to melt the corresponding amount of ice. This assumption
means that the melt water immediately attains the mixed-layer temperature.
We are interested in freshwater added or removed, and since the latent heat of
melting in sea ice decreases with increasing salinity (Defant, 1961), the e!ects
of freezing or melting on the salinity of the water column are largely independent
of the assumed salinity of the ice. S is therefore set to zero, and we introduce the
freshwater #uxes M"o o\M and F"o o\F and make the approximation
o "o "o"1000 kg m\.
U
In the present thought experiment the temperature of the water column is above
freezing. Since the controversy between Fridtjof Nansen and Otto Petersson at the
turn of the century (Gade, 1993), it has been commonly accepted that ice melting on
warmer water creates a less saline, colder and less dense surface layer, also if heat is
simultaneously lost to the atmosphere (Fig. 1). We assume that su$cient wind stirring
is present to homogenise this developing surface layer and that the entrainment from
Fig. 1. The evolution of the temperature and salinity pro"les as a mixed layer develops out of a
homogeneous warmer water column. The pro"les are scaled as to show the magnitude of the density
contributions.
1392
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
below can be described by a one-dimensional energy balance model. Keeping only
turbulent stirring and the surface buoyancy input, the mechanical energy budget
equation gives.
2m u
B
H !e
w "Max
;0
(2)
g* oH
g* o
"
"
for the entrainment velocity at the bottom of the turbulent mixed layer (Niiler and
Kraus, 1977). * "( ) !( ) is the di!erence in properties between the deep ocean
"
"
D and the mixed layer, u is the friction velocity and the "rst term on the r.h.s. of (2)
H
refers to stirring by the surface stress, m is a dissipation factor and we adopt the
laboratory value"1.25 (Kato and Phillips, 1969). The second term on the r.h.s. gives
the e!ects of the buoyancy #ux B at the sea surface. We apply e"1 for B*0, but
e"0.05 for B(0; i.e. free convection is not very e$cient in contributing to entrainment (Walin, 1993). In detrainment situations (2) gives w "0.
For a linear equation of state, the buoyancy step at the base of the mixed layer is
* o
g " "g* o"* b"g[b(S !S)!a(¹ !¹)],
"
"
"
"
o
(3)
where a and b are coe$cients of heat expansion and salt contraction, respectively. The
boundary conditions for temperature and salinity at the base of the mixed layer are:
"w (* ¹, * S).
X\&
"
"
Mass continuity for the mixed layer gives
w(¹, S)"
(4)
dH
"w #M, w *0;
dt
(5)
and with the approximations made above the temperature and salinity changes in the
mixed layer become
M¸
+(1!f )Q#fQ,
d¹
Q
H "! #
#w * ¹"!
#w * ¹,
"
"
c
oc
dt
oc
(6)
dS
H "!MS#w * S,
"
dt
(7)
where in (6) f is the fraction of heat going to ice melt and (1!f ) is the part transferred
to the atmosphere. According to our hypothesis f must attain such a value that the
buoyancy input;
gbwS"
fQ
"gbMS" gbS
8
o¸
(8)
is a minimum. Since o, ¸, b and S can be regarded as constants, it follows that fQ must
be a minimum. We assume that the f"f for which the minimum occurs is constant
for a given con"guration and proceed to determine f .
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1393
The density increase due to cooling is much smaller than the density decrease due to
added melt water. We therefore do not expect that cooling leads to convection or
generates any buoyancy-driven entrainment, when ice is present. Cooling, however,
reduces the buoyancy input, and the net e!ect of cooling and melting is considered in
the buoyancy #ux B the sea surface. When T is above freezing, B is given by
B"g
fc
fc
d(* ¹)
d(* ¹)
w * ¹bS# bSH "
! w a* ¹#Ha "
¸ "
"
¸
dt
dt
.
(9)
The "rst bracket on the right-hand side relates to the melting process, the second to
cooling. In each bracket the "rst term arises from the entrainment of warmer water
and the second from cooling the mixed layer.
The salinity S of the mixed layer develops according to
S R w dt
S" " ,
R (w #M)dt
which, using the "rst bracket in Eq. (9), may be written as
(10)
S R w dt
" S"
(11)
R w dt#R w fc/¸* ¹ dt#fc/¸R d(* ¹)/dtRYw dtdt
"
"
where the neglect of the heat capacity of the melt water allows us to write H"w dt
in the last term in the denominator. Integrating by parts then gives
S
"
S"
.
(1#fc¸\* ¹)
"
Introducing the expressions for B and S into (3) leads to
(12)
2m u
H
w"
S
"
g b S !
) !a* ¹ H
" (1#fc¸\* ¹
"
"
fc¸\S
d(* ¹)
d(* ¹)
"
g b
w * ¹#H "
!a w * ¹#H "
"
(1#fc¸\* ¹) "
dt
dt
"
!
S
"
g b S !
!a* ¹
" (1#fc¸\* ¹)
"
"
(13)
for the entrainment velocity, which simpli"es to
H d(* ¹)
m u (1#fc¸\* ¹)
H
"
" .
w"
!
(14)
g* ¹( fc¸\bS !a(1#fc¸\* ¹))H 2* ¹ dt
"
"
"
"
The deepening of the surface mixed layer combines the entrainment of sea water from
below and stirring of melt water from above according to
dH
fc
fc d(* ¹)
"w 1# * ¹ # H " .
"
dt
¸
¸
dt
(15)
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B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
When introducing Eq. (14), this gives
dH
m u (1#fc¸\* ¹)
H(1!fc¸\* ¹) d(* ¹)
H
"
"
" .
"
!
dt
dt
g* ¹( fc¸\bS !a(1#fc¸\* ¹))H
2* ¹
"
"
"
"
(16)
In the second term on the right hand side H is reduced by the added fraction of melt
water. As we neglect this contribution in the heat budget, we may write Eq. (16) as
d(H* ¹)
2m u (1#fc¸\* ¹)
" "
H
"
.
(17)
dt
g( fc¸\bS !a(1#fc¸\* ¹))
"
"
In the bracket (1#fc¸\* ¹) the term fc¸\* ¹ is small (+0.01). We therefore
"
"
neglect the time variation of * ¹ in (1#fc¸\* ¹). The initial mixed layer depth is
"
"
zero, and Eq. (17) integrates to
fc
H" 1# * ¹
¸ "
2m u t
H
.
(18)
g* ¹( fc¸\bS !a(1#fc¸\* ¹))
"
"
"
Eq. (14) for the entrainment velocity, using the same approximation of H as in the heat
budget, then becomes
m u
H
w"
g* ¹ *
"
(t d(* ¹)
1
1
"
!
*
2t (2* ¹ dt
(fc¸\bS !a(1#fc¸\* ¹)
"
"
"
(19)
and the heat loss is given by
Q" m u g
H
* ¹
t d(* ¹)
1
" #
"
.
2t
2* ¹ dt
(fc¸\bS !a(1#fc¸\* ¹)
"
"
"
(20)
Within our approximations only the last factor contains f, and the requirement of
minimum buoyancy input
d( fQ)/df"0
(21)
then leads to
2a¸
f "
.
(22)
c(bS !a*¹)
"
With values of a, b, * ¹ and S representative for the high-latitude ocean f +0.23.
"
"
That is, about of sensible heat lost by the ocean goes to ice melt. The rest is
transferred to the atmosphere. The constant expression validates the assumption that
f is constant used in the derivation above. However, a varies strongly with temperature, and a di!erent strati"cation with considerably warmer deep water would, while
not changing (26), lead to a larger value for f . f is twice the value required for
marginal stability at the base of the mixed layer, i.e. * o"0 for f" f and when
"
f) f no mixed layer can develop. The strati"cation would be unstable, and
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1395
convection a!ects the entire water column. It may be noted that ( f )\, neglecting
a *¹ , is equal to the stability parameter R introduced by Walin (1993). f also implies
"
that the contribution of melt water to the density step at the base of the mixed layer is
twice as large and opposite to the temperature contribution. The total loss of sensible
heat and the fractions going to the atmosphere and to ice melt are shown in Fig. 2 as
functions of f.
When substituting Eq. (22) into Eq. (14) the #ux of sensible heat at the sea surface
becomes
Q m u (bS #a* ¹) H d(* ¹)
"
" #
" H "
(23)
co ga(bS !a* ¹)H
2 dt
"
"
and since bS a *¹ this gives
"
Q m u H d(* ¹)
"
+ H#
(24)
2 dt
co gaH
Fig. 2. The total #ux of oceanic sensible heat, Q, the heat going to ice melt, fQ, and the atmosphere,
(1!f )Q, as functions of f for the case of a mixed layer formed out of ice melting on top of warmer water.
The heat loss is arbitrary. The value f giving minimum ice melt rate is indicated. For f" f the mixed
layer is marginally stable.
1396
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
The "rst term on the right-hand side gives the entrained heat Q , which then is
independent of the strati"cation and only depends upon H. This determines a smallest
possible depth H for a mixed layer formed by the melting of sea ice. For depth less
than H the second term in Eq. (24) is negative, which requires that the upper layer
becomes warmer. This is not possible because of heat loss and melting. The mixed
layer therefore instantaneously attains the depth H , where the entrained heat, Q ,
balances the heat loss to ice melt and the atmosphere. When the depth increases, the
mixed layer must cool to supply the heat loss to the atmosphere and to ice melt,
increasing the temperature step * ¹ (Fig. 1).
"
In the derivation of f it was implicitly assumed that the atmosphere absorbs all
heat lost by the ocean, and not going to ice melt. In reality this is not likely. The heat
#ux to the atmosphere is set by the temperature di!erence between the atmosphere
and the sea and upper ice surfaces (see Section 3 below). We therefore expect H to
adjust in such a way that the heat #ux to the atmosphere is balanced by entrainment.
This means that H must be di!erent for di!erent f . To brie#y examine the conse
quences of this, we assume that another hypothetical fraction f "r f goes to ice melt.
Eq. (22) then takes the form
2ra¸
f "rf "
c(bS !a *¹)
"
which introduced into (15) gives the expression
m u
H d(* ¹)
Q
H #
"
+
co ga(2r!1)H 2 dt
(22)
(24)
corresponding to Eq. (24). To supply same heat #ux to the atmosphere with f as with
f we must have
(1!f )Q"(1!f )Q
(25)
where Q is the total heat #ux in the primed system, The amount of heat going to ice
melt is then
(1!f )
Q
f Q"rf
(26)
(1!rf )
The e!ects of ice melt on the stability and entrainment are not considered here and f is, as expected, larger than f for r'1 and smaller for r(1. However, the melt water
has to be stirred into a mixed layer of depth H , where the heat loss to the
atmosphere balances the entrained heat. The buoyancy #ux multiplied by the depth
gives the change in potential energy as the initial mixed layer is formed. Since the
buoyancy #ux due to ice melt is proportional to the heat going to ice melt, this change
is proportional to
(1!f )mu H
H f QH "rf
(1!rf )ga(2r!1)H
(27)
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1397
where Eq. (24) and Eq. (26) have been used. The change of potential energy is
a function of the ratio r"f f \, and it has a minimum for r"(2f )\+1.47. This
corresponds to a value of 0.33 for the fraction f going to ice melt. This is di!erent
from f . Nothing else is to be expected considering the di!erent requirements for
which f and f were derived. What is interesting is that the initial mixed layer depth
may adjust in such a way that a minimum of the energy input, related to the stirring of
melt water into the mixed layer, does exist. It is also encouraging that the fraction for
which it occurs is not far from f , giving minimum ice melt, which was derived above
and which we shall adopt for the rest of this work.
A second observation about H is that it is not related to the Monin}Obukhov
depth, which gives the maximum depth a mixed layer, subject to wind mixing and
buoyancy input, can attain. H is the minimum depth a homogeneous mixed layer
with temperature above freezing and subject to cooling, melt water input and wind
stirring may have. The di!erent cases run in Section 4 show that, if its temperature is
above freezing, the mixed layer will attain a stage with no deepening, which corresponds to the Monin}Obukhov depth.
2.2. The evolution of the mixed layer in winter
The high-latitude mixed layers have "nite depth, are less saline than the underlying
water, and are cooled to freezing temperature in autumn. If a mixed layer with depth
(H#d)m is looked upon as Hm of water from the underlying ocean to which dm of
freshwater have been added, salt conservation gives the relation between S and S as
"
(H#d)S"HS
(28)
"
It is the freshwater content that controls the stability at the base of the mixed layer.
We therefore introduce the freshwater fraction f , which relates d to the sensible heat
that is made available, if Hm of underlying water is cooled by * ¹ degrees to attain
"
the mixed layer temperature. We then obtain
¸d
f "
c* ¹H
"
and eliminating d using Eq. (28) gives
(29)
¸(S !S)
f " "
.
(30)
(c* ¹)S
"
We shall use f instead of S when describing the mixed layer evolution. f enters the
expression for the buoyancy step below the mixed layer, Eq. (3), as
f cbS
"
g* o"ga* ¹
!1 .
(31)
"
"
a¸(1#( f c/¸)* ¹)
"
In a situation with no initial mixed layer as considered in Section 2.1, the f entering
the buoyancy #ux (Eq. (9)), and the f entering g*o (Eq. (31)) are the same. This is so
1398
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
because ice melt and cooling alone control the stability at the base of the mixed layer.
When an existing mixed layer is brought to freezing temperature and ice forms,
negative buoyancy input due to brine release is present, and this coupling is broken.
However, freezing does not exclude the possibility of melting, and the volumes of ice
formed and melted may be substantially larger than the net ice formation. We shall
require that the fraction of sensible heat going to ice melt, also in this situation, is
given by the f . We assume that the mixed layer must be at freezing temperature for ice
to form, and the expression for the entrainment velocity becomes
c
m u 1#f * ¹
H
¸ "
w"
cbS
c
"! 1#f * ¹
ga* ¹ ( f #f )
"
2a¸
¸ "
.
H
gbS
"
0.025(Q (1!f )Q )
o¸
#
cbS
c
"! 1#f * ¹
ga* ¹ ( f #f )
"
2a¸
¸ "
.
(32)
When ¹"¹ the cooling term disappears and the loss of sensible heat by the ocean,
D
Q, is equal to Q , and the heat #ux Q to the atmosphere now also includes latent heat
released by freezing. The positive buoyancy term due to ice melt has, as in Section 2.1,
been incorporated into the mechanical stirring term. The second term comes from the
negative buoyancy input due to the ice formation needed to close the heat balance
with the atmosphere. The appropriate values for e (1 and 0.05) for the two processes
have been introduced. The dynamic e!ect of freezing and brine release is kept separate
from the buoyancy input from melting. Freezing is unequivocally distinct from
melting, and the released brine has high enough density excess to convect in spite of
neighbouring areas with melting ice. It then contributes to the stirring of the mixed
layer and to entrainment of water from below.
2.3. Convection, ice melt and restratixcation
By freezing (and also by cooling, when no ice is present) small density anomalies,
positive as compared to the mixed layer, continuously form at the sea surface and
convect. The density di!erence between the mixed layer and the deep ocean disappears when f " f H. w are still "nite, and the convecting parcels will sink into the
deep ocean. The mixed layer is then removed and the underlying ocean rises toward
the sea surface and the ice, and a melting situation as the one discussed in Section 2.1
would follow. Convection and restrati"cation take place simultaneously. We assume
that convection only occurs in a small part of the area, but the convection velocity is
much larger than the entrainment velocity and the old, dense mixed layer is removed.
A new mixed layer then forms out of the by-passed, rising part of the underlying water
column (Fig. 3). Since it is created by melting, the results from 2.1 apply. A new
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1399
Fig. 3. The breakthrough of the convection out of the mixed layer and the reformed mixed layer after t"t
when the entrained heat balances the heat loss to the atmosphere.
mixed layer is considered established when it has attained the depth, H , where the
entrained heat balances the heat loss to the atmosphere (Fig. 3). From Eq. (24) we get
(1!f )m u oc
H .
H "
gaQ
(33)
In the situation considered in Section 2.1 no convection and removal of an old
mixed layer take place, and this depth was attained immediately. Here a phase exists
when water from below is entrained into the receding mixed layer. Not having
a convection model to tell us the time needed for the mixed layer to convect, we
instead estimate the time needed for a volume, corresponding to H , to become
entrained across the lower boundary. The entrainment velocity changes during this
phase because of the changing depth of the mixed layer. The mean entrainment
velocity is therefore, ad hoc, approximated by w H(H )\, where w is the entrainment
velocity and H the depth of the mixed layer immediately before the convection; the
ratio H(H )\ is introduced to take into account the increase in entrainment velocity
as the depth of the mixed layer decreases because of the convection. The time
t needed for the mixed layer to reform then becomes
t "H/w H.
(34)
Heat is supplied to the atmosphere during this restrati"cation phase, and once t is
determined, the temperature step * ¹ at the base of the newly formed mixed layer is
"
estimated from
Qt
* ¹"
.
(35)
"
(1!f )ocH
As will be noticed below, the adopted expression for t makes * ¹ about half the
"
previous temperature step, and after the convection a mixed layer with temperature
above freezing is formed. The salinity of the new mixed layer is close to that just before
1400
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
the convection, and the stable density step due to salinity is twice the unstable density
step due to temperature. This happens because restrati"cation occurs so quickly that
the atmosphere cannot absorb all sensible heat in the reformed mixed layer. The ice
melt is limited, since it is coupled to the heat going to the atmosphere, not to
a requirement that the mixed layer shall remain at freezing temperature. The ice melt
during the restrati"cation phase leads to reduction of the ice cover, and when the
mixed layer is established, the heat loss to the atmosphere will be larger than the
Q used to determine H .
As the mixed layer deepens, entrainment alone cannot supply enough heat to
balance the heat loss to the atmosphere, which also increases because of melting,
leading to thinner ice and larger fraction of open water. The heat stored in the mixed
layer is used and * ¹ increases as
"
d(* ¹) Q !(1!f )Q
" "
.
(36)
dt
(1!f )H
All fresh water comes from ice melt and f "f and the entrainment velocity is given
by
c
m u 1#f * ¹
H
¸ "
w"
cbS
c
"! 1#f * ¹
ga* ¹ ( f #f )
"
2a¸
¸ "
H d(* ¹)
" .
!
2* ¹ dt
"
H
(37)
If the temperature again falls to the freezing point, ice begins to form, and w is again
given by Eq. (32). The density step starts to decrease and is eventually removed and
a new deep convection event occurs. Should the ice cover disappear before freezing
temperature is reached, continued cooling makes the mixed layer unstable before ice
can form, or only a brief spell of freezing occurs before the layer convects. The ice then
melts, and if the stability of the underlying ocean is weak, no further ice formation
takes place. The rest of the winter will be dominated by thermal convection.
3. Heat loss to the atmosphere and evolution of the ice cover
So far it has not been necessary to consider explicitly the atmospheric forcing and
the evolution of the ice cover. However, to study concrete situations some parameterisation of these processes must be introduced. We mainly follow Washington
and Parkinson (1986), and for a more thorough treatment of sea ice mechanics and
thermodynamics we refer to this work and to Hibler (1979), Hibler and Flato (1992)
and the books edited by Untersteiner (1986) and Smith (1990).
The heat #ux to the atmosphere is approximated by a Newtonian transfer law
(Washington and Parkinson, 1986):
Q "K; (¹ !¹ )#¸
Q "K; (¹ !¹ )
(38)
(39)
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1401
Q speci"es the surface heat #ux for open water, with ¸ being the latent heat loss, Q is
the heat #ux over ice, and ; is the wind speed. It is assumed that the main e!ects of
the long wave radiation can be parameterised by a larger ¹ , and ¹ is an `e!ectivea
air temperature. Only the winter season is considered, and incoming short wave
radiation is neglected. ¹ and ¹ are surface temperatures for ocean and ice, respec
tively. The value of the transfer coe$cient K is kept constant. To quantify ¸ we
assume a constant di!erence of 0.002 between the speci"c humidity of the air, q , and
its corresponding saturation value, q . ¸ is then obtained from
¸ "K; (q !q )¸ /c ,
(40)
where ¸ "2.5;10 J kg\ gives the value for latent heat of vaporisation. The
friction velocity u is related to the wind velocity by the friction law
H
o
u " C ;.
(41)
H o "
The growth of the ice cover has two components: lateral and vertical. The lateral
growth reduces the area of open water. The vertical growth increases the ice thickness.
Following Parkinson and Washington (1979) we simulate the growth in both components by hypothesising that when freezing starts, the ice formed in the open water is
accumulated onto, initially in"nitesimally narrow, ice #oes of initial thickness d .
These #oes then grow vertically by heat conduction and freezing at their lower
boundary and horizontally by a continuous, equally distributed, accumulation at the
sides of the #oes, of ice formed in open water. The existence of motions to support
such a pattern, like ice convergences created by wind or currents, is assumed. In
melting phases the ice thickness d is not allowed to become smaller than d . As an
initial value d "0.1 m is used.
The heat budget for open water involves two sources, entrainment and freezing, to
balance the loss to the atmosphere. The total ice volume produced in open water is
given by I, and the volume production rate is
(Q !Q )A
.
IQ " (42)
o¸
We assume that the mixed layer must to be at the freezing point for ice to form. This
means that Eq. (40) only holds when Q"Q . Also in this case f Q of the oceanic
sensible heat loss goes to ice melt, but when the mixed layer water is at freezing
temperature, it is not necessary to separate melting and freezing in the ice balance. The
accumulation of ice at the edge of the existing ice #oes changes A, the fraction of open
water, according to
(Q !Q )A
.
AQ " (43)
o ¸d
We only study seasonal ice covers formed in the Greenland Sea and the Weddell
Sea, and their thickness will be limited to about 0.5 m. We therefore do not consider it
necessary to apply a complex multi-layer ice model, including snow cover, and we
1402
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
assume that the ice can be approximated by a homogeneous, conducting slab. The
heat loss through the ice is then given by
Q "i (¹ !¹ )/d
(44)
where the lower ice boundary is assumed to be at freezing temperature. The heat
transfer between water and ice is not explicitly modelled. When ¹'¹ entrainment
and cooling of the mixed layer supply all heat both to ice melt and to the atmosphere.
For ¹"¹ entrained heat goes both to ice melt, and is conducted through the ice.
Freezing at the bottom then closes the balance with Q . By substituting Q from (39)
the ice surface temperature is eliminated (Hibler and Flato, 1992);
i K;
Q" (¹ !¹ ).
(45)
i #K; d
The coe$cient of heat conduction in ice, i "2.0 J 3C\ m\ s\, and the coe$cient
of heat transfer, K"1.43 J 3C\ m\, keep their adopted values throughout. If the
storage of sensible heat in the ice is neglected, the vertical growth of the ice is given by
dQ "(Q !Q)/o ¸.
(46)
For simplicity the salinity of sea ice is again set to zero. The salt content of sea ice is
due to trapping of sea water between ice crystals in the freezing process. This water
will increase the thickness of the ice. The overestimate of the heat conduction through
the ice cover caused by underestimating its thickness is not expected to be serious.
In melting phases the heat loss to the atmosphere is given by
Q "AQ #(1!A)Q "(1!f )Q
(47)
where the right-hand side gives the loss of sensible heat by the ocean to the atmosphere. The temperature at the lower boundary of the ice is kept at the freezing point
and Q is still given by (45). We use this combined equation because when T is above
freezing no new ice forms in open water and it is not be possible to separately balance
the heat loss through the ice and the heat loss in the open areas. The change in ice
volume v becomes
v "f Q/o ¸.
We approximate the fraction of open water by an exponential function:
A"exp+!jv,
(48)
(49)
where the scale factor
ln A
j"
(50)
v
is set from A , the area of open water, and v , the ice volume, just before the
convection. The thickness of the ice then evolves as
dQ "v/(1!A).
(51)
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1403
This is prescribed to hold until the ice thickness is reduced to d . After that the ice is
considered to be predominantly collected into ice bands of thickness d . The expres
sions for the changes in ice volume and area of open water are then given by
v "d
d(1!A)
f Q
f Q
"! N AQ " .
M dt
o¸
d o¸
(52)
4. Numerical examples
The present integral approach describes the evolution of bulk parameters such as
mixed layer salinity and depth, the heat #uxes, the entrainment velocity and the
average ice thickness and the ice surface temperature. It should then answer the
question: Is deep penetrating haline convection in the ocean possible, at least to
the extent that the present formulation of the problem is valid?
The model is one-dimensional and has two layers, a mixed layer above a deep
homogeneous ocean. Computations are made with 1-hour time steps. The mixed layer
is initially at the freezing point and w is determined using Eq. (31), which multiplied
by the temperature step gives the heat #ux from the lower layer. The heat transfer to
the atmosphere at open water and through ice is given by Eqs. (38) and (39).The heat
conduction through the ice is then found from Eq. (45) with an initial ice thickness d .
The increase in ice thickness during one time step is obtained from the di!erence
between entrained heat and heat conduction. The thickness multiplied by the fraction
of ice covered water then gives the ice volume formed under ice. The change in the
area of open water is taken from Eq. (42), where the ice thickness from the previous
time step is used. The ice growth in open water is then computed by the change in area
of open water times the ice thickness.
Entrainment and ice formation change the depth of the mixed layer as
o
dH
"w ! (IQ #(1!A)dQ )
o
dt
and its salinity by
d(HS)
"w S
"
dt
(53)
(54)
If the stability (b* S!a* ¹) at the base of the mixed layer becomes zero, the mixed
"
"
layer convects. The equilibrium depth of the new mixed layer and the time needed for
it to be established are estimated from (33) and (34). The new * ¹ is determined from
"
(35) and, because the new mixed layer is formed from melting ice, f "f . For ¹'¹
the entrainment velocity is computed using (37). The heat loss and the reduction of the
ice cover are calculated using (47)}(49). H and S are again found using (53) and (54) and
the temperature change is determined from
d¹ Q !Q
"
.
dt
H
(55)
1404
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
If d becomes equal to d the ice thickness stays constant and the area of open water
changes as (52). If T reaches the freezing point before all ice is melted, ice starts to form
and the initial expressions are adopted. A second instability arises, deep convection
follows and again a mixed layer is reformed. The calculations were continued until the
ice cover had disappeared. In cases with high initial stability and no convection the
computations were terminated after 208 days.
4.1. The relative ewects of wind speed, air- and deep water temperature
A mixed layer initially at freezing temperature, with salinity S"34.65 and a depth
of 80 m lies above a deep layer with ¹ "!0.93C and S "34.85. The atmospheric
"
"
forcing is given by ; "5 or 10 m s\ and ¹ "!20 or !303C. The time evolution
of di!erent parameters are shown in Fig. 4 for ; "5 and 10 m s\ and
¹ "!303C. The "rst panel displays the variations of the heat #ux to the atmosphere, Q , and the heat entrained from below, Q . The second panel shows the
atmosphere and ice surface temperatures ¹ and ¹ . The third panel indicates the
changes in ice thickness d, and fraction of open water A. Panel 4 gives the mixed layer
depth H, panel 5 the temperature in the mixed layer and the underlying ocean T,
¹ and panel 6 the corresponding salinities S, S . The initial conditions are given in
"
"
Table 1 and the results are summarised in Table 2.
The greatest di!erence arises from changes in wind speed. When the wind is strong,
Q and Q almost balance (Fig. 4a, "rst panel). The ice initially grows rapidly and thin
ice covers the entire sea surface (3rd panel). There is a weak increase in the heat
transfer from below (1st panel), and the ice thickness has a maximum and then begins
to melt slightly (3rd panel). The mixed-layer depth increases rapidly as shown in
panel 4. After the initial net ice growth, because of the established balance between
ice formation and ice melt, the salinity increase is due to entrainment from below
(6th panel). The entrained water loses its heat but adds salt to the mixed layer. The
density of the mixed layer increases until the stability is gone. The mixed layer then
convects. In the case with lower temperature just enough ice is formed to create a new
mixed layer of depth H after the convection. (The gaps in the diagrams indicate the
unresolved periods of convection and restrati"cation before the formation of a mixed
layer with depth H .) Less than 3 days after the new mixed layer is established, the ice
has melted. With the higher air temperature the ice thickness is always less than the
initially prescribed d "0.1 m and not enough ice is formed to create a mixed layer of
thickness H after the convection (Table 2).
With a lower wind speed the picture is di!erent (Fig. 4b). No balance is reached
between Q and Q before the stability is gone and the mixed layer convects (1st panel).
A small fraction of open water is present and the ice thickness is still increasing, when
the stability is gone (3rd panel). After the convection a mixed layer with temperature
above freezing is created. As the mixed layer deepens, its temperature and salinity is
reduced, but f stays constant. The larger layer depth and stability reduce the
entrainment velocity, and it becomes zero. The mixed layer has now attained the
Monin}Obukhov depth, which is about twice as deep as H (4th panel). A cooling, but
no deepening, of the mixed layer then follows until the mixed layer temperature
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1405
Fig. 4. The evolution of the mixed layer. Panel 1 shows the heat loss to atmosphere, Q , and the entrained
heat, Q (W m\); Panel 2 the atmosphere and the ice surface temperature, ¹ and ¹ (3C); Panel 3 the ice
thickness, d (m) and the fraction of open water, A. Panel 4 gives the mixed layer thickness, H (m); Panel
5 shows the mixed layer temperature, T and the deep water temperature ¹ (3C) and panel 6 the
"
corresponding salinities S and S . The values are given for every 12 hour before the "rst convection and
"
then after every 6 hour. The gaps indicate the unresolved convection and restrati"cation periods t . The
stability across the base of the mixed layer is the same for the two cases and the temperature and salinity of
the lower layer are !0.93C and 34.85. Air temperature is !303C. (a) ; "10 m s\, (b) ; "5 m s\.
1406
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
Fig. 4. Continued.
reaches the freezing point and ice begins to form (3rd and 5th panels). The stabilising
e!ect of net melting then disappears, and the mixed layer again deepens. The heat loss
to the atmosphere is reduced because of increased ice thickness and smaller A
(3rd panel). The decrease in stability caused by freezing allows the entrainment from
below to become large enough to balance the heat loss to the atmosphere (1st panel).
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1407
Table 1
Initial conditions for the 6 test cases
exp.
¹ (3C)
S
¹ (3C)
S
H (m)
¹ (3C)
; (ms\)
1
2
3
4
5
6
!1.9
!1.9
!1.9
!1.9
!1.9
!1.9
34.65
34.65
34.65
34.65
34.65
34.65
!0.9
!0.9
!0.9
!0.9
!0.5
!1.3
34.85
34.85
34.85
34.85
34.87
34.83
80
80
80
80
80
80
!30
!20
!30
!20
!25
!25
10
10
5
5
7
7
No net ice formation then occurs. The salinity continues to increase because of
entrainment, and f drops toward f . The mixed layer again becomes unstable and
convects. After the convection the mixed layer is shallower than the initially prescribed 80 m, but it more than doubles its thickness before the next convection event
occurs. For ¹ "!203C the mixed layer is reformed 4 times after which the ice
disappears. When ¹ "!303C enough ice has formed for 6 deep convection events
to occur. The time intervals between the convection events decrease because freshwater is exported by convection into the deep ocean and the ice thickness is reduced,
causing a larger heat loss to the atmosphere. However, the fraction of open water does
not rise above 0.2 until at the very close to the disappearance of the ice (3rd panel).
The e!ects of the temperature di!erence between the mixed layer and the underlying water are studied in cases 5 and 6 (Table 1 and 2). The same forcing (; "7 m
s\ and ¹ "!253C) is applied to a mixed layer situated above deep layers with
temperature ¹ "!0.53C and ¹ "!1.33C respectively. The initial mixed layer
"
"
depth is 80 m both cases. A higher temperature in the underlying water leads to larger
Q and thinner ice cover. In both cases a balance between Q and Q is reached before
the water column becomes unstable. The convection starts earlier for the colder water
column, while the ice disappears "rst for the warmer deep layer. Case 6 with larger ice
volume and lower deep water temperature requires three convection events before the
ice cover has melted. For the warmer second layer (case 5) only one deep convection
occurs (Table 2).
From these idealised situations it appears that cold air, weaker winds and cold
underlying water are the conditions most favourable for generating instability and
deep convection events. Strong winds and perhaps also higher temperatures in the
second layer promote a larger heat loss from the ocean and a more rapid removal of
the ice. In a model formulated like the present one, deep convection has to occur at
least once, before the ice can be removed by melting.
Several cases also were run with the same forcing as 5 and 6 but with di!erent
f (Table 3). The convection starts earlier for larger f. The larger fraction of heat used to
melt ice increases the stability at the base of the mixed layer, and less heat is entrained.
This leads to stronger ice formation. More brine is released into a shallower layer, and
the water column becomes unstable more rapidly in spite of a thicker ice cover. For
small f the stability at the base of the mixed layer is weaker and entrainment
36
52
17
36
45
34
1
2
3
4
5
6
0.12
0.05
0.35
0.32
0.20
0.27
d (m)
!1.9
!1.9
!1.9
!1.9
!1.9
!1.9
¹ (3C)
34.80
34.80
34.80
34.80
34.80
34.80
S
238
278
99
130
155
181
H (m)
!1.33
*
!1.3
!1.32
!1.1
!1.56
¹ (3C)
34.81
*
34.81
34.81
34.81
34.80
S 82
*
25
36
53
64
H (m)
1
1
6
4
1
3
N
*44
*
86
130
61
75
t (d)
!1.57*
*
!1.63
!1.83
!1.60
!1.86
¹ (3C)
34.79
*
34.78
34.76
34.76
34.77
S 91
*
17
27
56
44
H (m)
Note: t is the time to the "rst convection and t is the time required for melting the ice. N is the number of convection events. In case 2 not enough ice is present to
form a new mixed layer. All other symbols are as in the text.
Just before the "rst convection.
Just after the "rst convection.
After the melting.
t (d)
Exp.
Table 2
The characteristics of the mixed layer
1408
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1409
Table 3
Experiments 5 and 6 with di!erent values of f
Exp
f
EB
t (d)
d (m)
H (m)
N
t (d)
5
5
5
5
5
5
5
6
6
6
6
6
6
6
0.80
0.50
0.25
0.23
0.21
0.19
0.16
0.80
0.50
0.25
0.23
0.21
0.19
0.16
!
!
!
#
#
#
#
!
!
!
#
#
#
#
13
17
40
45
50
53
66
13
15
29
34
39
46
60
0.36
0.34
0.23
0.20
0,17
0.12
0.03
0.36
0.35
0.29
0.27
0.25
0.22
0.14
87
94
142
155
170
190
233
93
104
162
181
207
242
327
1
1
1
1
1
1
1
2
2
4
3
2
2
1
13
17
58
61
63
67
80
17
27
76
78
82
78
73
Note: EB indicates the energy balance of the mixed layer just before the "rst convection. # means that the
entrained heat is larger than the heat loss to the atmosphere. d and H are the ice thickness and the mixed
layer depth at the time of the "rst convection.
eventually supplies more heat than the atmosphere requires and net ice melt starts.
The mixed layer then reaches large depths and needs more time before it becomes
unstable and convects. Convection occurs for all cases with constant f, and the ice
starts to melt. When f is large the ice has melted before a mixed layer of depth H has
formed. For f around 0.25 several convection events occur before the ice is removed.
Q becomes slightly larger than Q for f"0.23 (our adopted value), but an approxim
ate balance between heat loss to the atmosphere and entrained heat is reached and no
net ice melt or freezing occurs when f"f . For f less than 0.20 only little ice is formed
and after the convection the ice quickly disappears.
4.2. The Weddell Sea
The importance of the Weddell Sea as a source of deep water for the world ocean is
well known. Most of the dense-water formation occurs close to the Antarctic continent, on the shelves, and under the #oating ice shelves (Gill, 1973; Killworth, 1983).
However, Gordon observed a cold, narrow and almost homogeneous 4000 m deepwater column close to Maud Rise in the central Weddell Sea gyre, which he interpreted as remnants of deep-reaching open-ocean convection (Gordon, 1978). This
convective feature or `chimneya was believed related to recurrent polynyas observed
from satellites in late winter during the mid-70s (Zwally et al., 1983). This discovery led
to intensive observational and modelling work to describe deep convection and
conditions necessary for polynya formation (Killworth, 1979; Martinson et al., 1981;
Martinson, 1990; Gordon and Huber, 1990; Lemke et al., 1991). We shall examine
conditions representative of the Weddell Sea. Two background strati"cations are
1410
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
considered: one based on observations in the `warma regime and the other using
observations at Maud Rise (Gordon and Huber, 1990). In the warm regime the
underlying water has a temperature of 1.33C and a salinity of 34.7. The salinity of the
upper layer in autumn is estimated to be 34.11 (Gordon and Huber, 1990). Four
situations with mixed-layer depths 20, 40, 60, and 80 m are considered. Air temperatures of !203C and wind velocities of 5 m s\ are assumed. Two additional cases,
one with an initial salinity of 34.21, the other with an air temperature of !253C are
also discussed. In these two cases the mixed-layer depth is set to 80 m. The results are
summarised in Table 4, and the evolution of the initially 80 m deep mixed layer for the
standard forcing is shown in Fig. 5. Table 4 also gives mixed-layer depths, entrained
heat and ice thickness as reported by Gordon and Huber (1990).
For initial depths of 20 and 40 m the density di!erence between the mixed layer and
the underlying water column disappears. The mixed layer convects and the ice melts.
For the thinnest layer not enough ice was formed to reform a mixed layer with depth
H . The 40 m thick layer convected after 137 days and the ice was gone after 175 days,
a month later than the end of the estimated winter period of 150 days (Gordon and
Huber, 1990). Such mixed-layer depths are, however, considerably shallower than
those observed in the area. In the other cases the water column remained stable
throughout the winter. The heat loss to the atmosphere and the entrained heat were
almost in balance by the end of the winter, and the heat loss at the sea surface was
reduced from 165 to 40}50 W m\. The mixed-layer deepening ranged between 25
and 35 m, and the ice thickness was between 45 and 75 cm. The salinity of the mixed
layer after the winter was between 34.49 and 34.52. These values are all in the range of
those reported by Gordon and Huber (1990). The area of open water had decreased to
Table 4
The evolution of the mixed layer and its properties after 150 days in the warm regime in the Weddell Sea for
di!erent initial and boundary conditions
H
¹ (3C)
S
t (d)
t (d)
d (m)
H (m)
Q
(W m\)
Q
(W m\)
S
20
40
60
80
80
80
!20
!20
!20
!20
!25
!20
34.11
34.11
34.11
34.11
34.11
34.21
59
129
*
*
*
*
163
*
*
*
*
0.06
0.25
0.45
0.63
0.73
0.50
60
96
106
115
119
122
109
69
50
39
45
46
108
69
50
40
46
46
34.54
34.54
34.505
34.492
34.505
34.515
*
*
0.55
111
41
41
Gordon and Huber 1990
68
!13
34.11
34.4 } 5
Note: The wind velocity is in all cases 5 m s\. The symbols are as in the main text and in Table 2. The
properties estimated by Gordon and Huber (1990) are also shown.
Not enough ice was formed to recreate a mixed layer of depth H .
The conditions just before the convection.
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1411
Fig. 5. The same as for Fig. 5, but with properties representative of the `warma regime in the Weddell Sea;
¹"!1.93C, S"34.11, ¹ "1.33C, S "34.7, ; "5 m s\, ¹ "!203C, H"80 m.
"
"
zero in all non-convecting cases in contrast to the about 5% open water that has been
observed in the region throughout the winter (Gordon and Huber, 1990).
Gordon and Huber (1990) estimated the amount of water entrained into the mixed
layer from observations of oxygen saturation and used this to quantify the oceanic
heat #ux to the mixed layer and the subsequent heat loss to the atmosphere. They
1412
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
applied an empirical expression for the heat loss through the ice to compute the air
temperature needed to conduct the heat through the ice with no net melting. They
found this temperature to be !133C, which agreed with the mean observed air
temperature. The ¹ we use is an `e!ectivea air temperature, which also includes the
radiation heat loss, and we should compare the !133C found by Gordon and Huber
with the ice surface temperature ¹ . This is about 63C higher than ¹ and for
¹ "!203C ¹ !143C (Fig. 5, 2nd panel).
At Maud Rise the second layer is considerably colder (!0.53C) and less saline
(34.6). Also here 6 di!erent situations were examined. Four cases had a mixed-layer
depth of 80 m, mixed-layer salinities of 34.21 and 34.31, and air temperatures of
!203C and !253C. In the last two cases the mixed-layer depth was 120 m, the
salinity 34.31 and the air temperatures !20 and !253C. The stability in all cases but
one was removed by the end of the winter (150 days) and convection did occur
(Table 5). However, this happened late in winter, and no polynya would have time to
develop before the summer ice melt. The increase in mixed-layer depth up to the "rst
convection event was between 25 and 50 m. The maximum ice thickness, occurring
just before the convection, was between 40 and 75 cm and the entrained heat between
35 and 60 W m\ (Table 5). No balance between heat loss to the atmosphere and
entrained heat was reached before the convection, and net ice formation still took
place.
For the shallow, high salinity case with the smallest heat loss the "rst instability
occurred after 80 days and the mixed layer was reformed 18 days later (Fig. 6).
A second convection event took place after 162 days, and again a mixed layer
was reformed. After a third convection event the ice "nally disappeared after 205 days
and a polynya was created. For the case with a higher heat loss, four convective
events took place before the ice had disappeared. This occurred after 171 days, slightly
longer than the estimated winter duration. The fraction of open water exceeded 50%
between the last two convection events and a polynya could be considered formed
before the last convection. These results indicate that deep convection may occur
at Maud Rise, but to create a polynya the initial salinity of the mixed layer must be
high.
Table 5
The evolution of the mixed layer at Maud Rise for di!erent initial and boundary conditions
H (m)
¹ (3C)
S
t (d)
d (m)
H
Q
(W m\)
Q
(W m\)
S
N
t (d)
80
80
80
80
120
120
!20
!25
!20
!25
!20
!25
34.21
34.21
34.31
34.31
34.31
34.31
160
101
80
50
137
87
0.61
0.66
0.43
0.47
0.68
0.72
164
139
132
116
183
169
41
48
51
59
37
42
41
49
51
62
38
46
34.53
34.53
34.53
34.53
34.53
34.53
4
5
3
4
4
3
394
313
205
171
406
324
Note: The wind velocity is 5 m s\ in all cases. The symbols are as in the main text and in Table 2.
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1413
Fig. 6. The same as Fig. 5, but with properties representative of Maud Rise, S"34.31, ¹"!1.93C,
S "34.6, ¹ "!0.53C, ; "5 m s\, ¹ "!253C, H"80 m.
"
"
4.3. The Greenland Sea
The #}S characteristics of the Greenland Sea water column show that a renewal of
the bottom water can occur through local convection. Several deep (1000}2000 m)
convection events have been observed in recent years (GSP group, 1990), but no
1414
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
convection reaching the bottom has been documented in the time period 1982}1996.
The water column in the Greenland Sea is also in#uenced by waters advected from the
Arctic Ocean and by Atlantic Water recirculating in Fram Strait and returning along
the Greenland continental slope. The Atlantic Water lies immediately below the low
salinity mixed layer. It is prominent closer to the rim, colder and often absent in the
centre. The Arctic Ocean out#ow supplies both ice and low-salinity surface water as
well as deep waters. The deep waters of the Arctic Ocean are warmer and more saline
than the deep waters formed in the Greenland Sea.
Roach et al. (1993) reported temperature, salinity and current measurements from
instruments moored for one year in the central Greenland Sea and in `Isoddena north
of Jan Mayen. They found that in the central Greenland Sea the mixed layer, after it
had reached freezing temperature and ice was forming, convected after about
1 months. Two to three weeks later the ice cover had disappeared. The interpretation
given by Roach et al. (1993) was that the density of the upper layer in the central gyre
had increased due to released brine, and that it convected into the deeper layers.
Warmer water was brought close to the surface, melting the ice. The new low-salinity
layer was then assumed advected from the central gyre. Deep, thermal convection was
established and continued for the rest of the winter.
Visbeck (1993) and Visbeck et al. (1995) studied the same area and period using
observations from moored thermistor chains and ADCPs, and applied a one-dimensional mixed-layer model forced by real meteorological data. To reproduce the
observed changes in the mixed layer they found that a net export of 0.005 m d\ of ice
out of the area was required, about 1/3 of ice production estimated by Roach et al.
(1993). From model calculations Visbeck et al. (1995) concluded that the entrainment
from below became so strong that the ice started to melt before the mixed layer
convected. After the ice cover was removed a deep thermal convection was established.
We shall examine two situations, where the second layer is dominated by Atlantic
and Arctic waters (¹ "!0.43C and ¹ "!1.43C), respectively. The initial salini"
"
ties of the mixed layer are set to 34.656 and 34.631, and ¹ "!253C and
; "7 m s\ are used for the atmospheric forcing. The salinities of the second layer
are 34.88 and 34.855, respectively, making the strati"cation in the Arctic situation
about 30% more stable. Until now only the e!ects of the characteristics of underlying
waters on the ice cover and the mixed layer evolution have been considered. However,
the second layer has a "nite thickness, which may be more important in the Greenland
Sea, with several sources of intermediate waters, than in the Weddell Sea. We
therefore also discuss the e!ects of the convection events on the second layer. Based
upon the results of the Greenland Sea Project (GSP group, 1990; Roach et al., 1993;
Schott et al., 1993; Visbeck et al., 1995) it is assumed that the two upper layers together
䉴
Fig. 7. The same as for Fig. 5, but with a "nite depth H of the second layer and properties representative of
"
the Greenland Sea. The evolution of the second layer salinity, S and temperature, ¹ (3C) is also shown
"
"
(panels 5 and 6). (H#H )"600 m, ; "7 m s\, ¹ "!253C. (a) ¹"!1.93C, S"34.656,
"
*o"0.119 kg m\, ¹ "!0.43C, S "34.88, f "0.345. (b) ¹"!1.93C, S"34.631, *o"0.159 kg
"
"
m\, ¹ "!1.43C, S "34.855, f "1.035.
"
"
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1415
make up 600 m of the water column, and we follow the evolution of the second layer
as well as the mixed layer. The mixed layer convects when it reaches marginal
stability. After a new mixed layer of depth H has formed, it is assumed that the
convected water mixes completely with the remaining part of the second layer. The
second layer becomes colder and fresher but retains its density. In the reformed mixed
layer f will now be larger than f because of the reduced temperature of the
1416
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
Fig. 7. Continued.
underlying water, and the changes in the second layer will a!ect the mixed-layer
evolution.
The two cases are shown in Fig. 7a and b. The "rst instability and convection arise
after about the same time, but a balance between heat loss to the atmosphere and
entrained heat has then been reached for the `Atlantica case, while there is still a net
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1417
heat loss in the low-salinity colder, `Arctica case (1st panels). After the convection the
ice rapidly melts in the Atlantic situation and a mixed layer with temperatures above
freezing remains. Apart from a slightly higher initial salinity this run recaptures the
observations by Roach et al. (1993), and the evolution of the temperature and salinity
of the two layers can be compared with those described by Roach et al. (1993). The ice
formation is smaller than that estimated by Roach et al. (1993) because a large part of
the salinity increase is due to entrainment of saline water from below. The freshwater
input from melting after the convection is then smaller, and a local freshwater balance
without export is conceivable. The freshwater is convected into the deeper layers.
In the second situation a series of convection events arises, occurring faster and
faster (Fig. 7b). The heat #ux from below does not go to zero between convection
events, but remains at a low and constant level for a longer period before ice formation
again starts (1st panel). The deepening between the convection events proceeds
rapidly because of the ever weaker stability, and a mixed layer thicker than 200 m
convects each time (4th panel). The ice thickness is reduced, but the ice has not
disappeared before the density di!erence between the two layers is removed by the
repeated injections of cold, low salinity water into the lower layer. Haline convection
would then reach the base of the second layer.
Two more cases will be discussed (Fig. 8a and b). The temperature of the second
layer is !1.33C, but the total depths are 400 and 600 m, respectively. The wind is
reduced to ; "6 m s\, and the initial stability is weaker than in the previous
situations. For the deeper second layer, four convective events occur before the ice is
removed. For the shallower second layer seven convective events take place, after
which the stability at the mixed layer base has practically disappeared. Ice however, is
still present. A shallower second layer becomes colder by convection, and a longer
phase of deep reaching haline convection is promoted.
A low-salinity, cold (but above freezing) mixed layer was observed in the winter
1987}88 after what was interpreted as a haline convection event (Rudels et al., 1989).
The temperature of the mixed layer was between that of the lower layer and the
freezing point, and the density step due to salinity was about twice that due to
temperature. This suggests a removal of the mixed layer into the deep and the rising of
the lower layer to the surface, leading to ice melt and to the reformation of a low
salinity mixed layer. In 1988}89 the absence of a cold, low salinity surface layer (GSP
group, 1990) indicates that the "nal stage of the convection was thermal as suggested
by Roach et al. (1993) and Visbeck et al. (1995).
Haline convection in the Greenland Sea is then mostly con"ned to the upper part of
the water column. Spells of deep reaching, haline convection are likely to be brief,
especially if Atlantic Water dominates in the second layer. The ventilation of the
deeper waters will then be caused by later, thermal convection and a deepening of the
mixed layer. When the Arctic component is strongly represented, the freshwater
content is large as well as the temperature low, and a situation with several convective
events arises. If the second layer is cold and thin, the two layers may merge during
active haline convection. The combined layer will then increase in density until still
deeper convection occurs. Most likely the ice will eventually disappear also in this
situation.
1418
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
Fig. 8. The same as in Fig. 7. ¹"!1.93C, S"34.70, *o"0.116 kg m\, ¹ "!1.33C, S "34.875,
"
"
f "0.669, ; "6 m s\, ¹ "!253C. (a) (H#H )"600 m, (b) (H#H )"400 m.
"
"
5. Summary
The present work examines whether deep convection, driven by brine rejection, is
possible in the ocean. It is found that within the context of energy balance mixed-layer
models, this depends upon how the loss of oceanic sensible heat is distributed between
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1419
Fig. 8. Continued.
atmosphere and ice melt. If a constant fraction goes to ice melt, or if the partitioning
has a minimum value larger than f /2, the stability at the base of the mixed layer will
disappear and the mixed layer convect before the ice is melted.
Is this realistic? This assumption could be questionable, when the stability is weak
and the ice cover thin and large areas of open water are present. This is also when the
1420
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
model studies by Walin (1993) and Visbeck et al. (1995) indicate the removal of the ice.
Because the stability of the mixed layer decreases with ice formation, it is expected that
the heat #ux from below, which depends inversely on the stability, eventually provides
more heat than is required by the atmosphere. However, melting increases the
stability and lowers the entrainment rate, and freezing may again be needed to supply
enough heat to the atmosphere, thus pushing the system toward renewed freezing.
Therefore, assuming that melting occurs, what is the maximum stability, or the
maximum freshwater content, the mixed layer must have to unequivocally sustain
melting?
If we write the entrainment equation in the form used by Walin (1993), taking into
account a net buoyancy input due to melting we get
(Q !Q )c¸\bS aQ
g
! oc
2m u
oc
H
w"
!
.
g(b* S!a* ¹)H
g(b* S!a* ¹)
"
"
"
"
The heat available for melting can be written as
(56)
Q
(Q !Q )"w co* ¹ 1!
.
(57)
"
w co* ¹
"
Inserting this into the expression for the entrainment velocity given above and making
the same rearrangement as in Section 2 we get
c
m u 1#f * ¹
H
¸ "
.
(58)
w"
Q
cbS
c
"
ga* ¹
1!
#f
! 1#f * ¹ H
"
2a¸
¸ "
w co* ¹
"
If we now solve for w and introduce * ¹, the expression for the entrained heat #ux
"
becomes
Q
w * ¹" "
oc
c
m u 1#f * ¹
H
¸ "
"
cbS
c
"! 1#f * ¹
ga (1#f )
2a¸
¸ "
H
Q
#
.
(59)
2a¸
c
oc 1# f !
1#f * ¹
cbS
¸ "
"
The bracket in the denominator of the second term is (0, if f (f . This means that
the heat #ux from below must be larger than the heat loss to the atmosphere, if the
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
1421
freshwater content in the mixed layer is so small that the stabilising density step due to
salinity is less than twice the destabilising density step due to temperature. To obtain
this we do not have to consider the "rst term as long as it is positive. This is in
agreement with the test cases considered in Section 4, where smaller fractions than
f were examined. It also agrees with the results obtained by Walin (1993).
However, to deduce that deep convection is not possible also requires the assumption that no ice is formed at all once the melting stage is reached. This is far more
di$cult to prove, theoretically as well as observationally, given all possibilities of how
the atmosphere-ice-mixed layer-ocean system may interact. One could argue that
a large amount of open water would make it easier for the entrained heat to avoid
melting ice and escape to the atmosphere. On the other hand, open water will favour
new ice production, and even if the ice is rapidly melted, a larger portion of the
entrained heat then goes to ice melt than the di!erence between entrained heat and
heat lost to the atmosphere indicates. The buoyancy input due to this added ice melt
would inhibit the entrainment enough to allow convection to start before the ice cover
is removed.
We have assumed that a preferred fraction f of sensible heat going to ice melt exists,
and have examined if this assumption leads to obvious contradictions with observational facts. This does not appear to be the case. We believe that the value obtained for
f is of the right order (see discussion in Section 2.1). The description of entrainment
and ice cover has, on purpose, been kept simple (one could even say retrograde, as was
pointed out by one reviewer). This is partly not to obscure our main goal, to study the
e!ects of the chosen distribution of the heat #ux. It also eases the comparison with the
studies by Walin (1993) and Visbeck et al. (1995). Many of the possible improvements
of the energy balance mixed-layer model, which can be made, would actually favour
freezing rather than entrainment and melting. Another assumption is that wind-mixed
entrainment is a larger obstacle to deep, haline convection than the entrainment
caused by the convection itself. This is the commonly accepted view, but it could be
wrong, especially if the mixed layer is deep. The model by Backhaus (1995) does imply
that brine-driven, haline convection may shut itself o!.
If deep haline convection does occur, a goal would be to combine a more sophisticated mixed layer description with a plume convection model. The results from the
situations studied in the Greenland Sea (Section 4.3) indicate that recurrent convective
events transform the second layer, making it colder and less saline. In that description
the convecting water was not denser than that of the second layer and its density did
not increase. In reality convective plumes must be slightly denser, and the cooling of
the second layer will, because of the higher compressibility of colder water, also reduce
the density step at the lower boundary of the second layer. In the situation with
remaining ice and haline convection breaking through the mixed layer, the plumes
may continue through the second layer. Haline plumes start with temperature at the
freezing point, and because only cold water is entrained in the mixed layer as well as
the second layer, the thermobaric e!ect on the plumes will be strong, when they reach
the lower boundary of the second layer. They may then have a large enough positive
density anomaly to sink into the deep ocean, perhaps by-passing some levels and
reach the bottom layer. In the case of thermal convection the initial density anomalies
1422
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
are smaller, the temperature higher and the thermobaric e!ect at the lower boundary
will be smaller, even if a deep ('2000 m) mixed layer exists as in 1989 (GSP-group,
1990).
These are, at present, idle speculations that require more thorough investigations.
However, this work is already long enough and introduces hypotheses that may
appear counter intuitive. It is therefore not the place to introduce another, controversial, subject, and we end the present study here.
Acknowledgements
We wish to thank Norbert Verch for technical assistance. This work has partly been
supported by the European Commission MAST II programme ESOP-I (MAS2CT93-0057). Economic support for one of us (BR) has been received from the
Deutsche Forschungsgemeinschaft (DFG-SFB-18, TP B3) and from the European
Commission MAST III Programme VEINS, through contract MAS3-CT96-0070.
Appendix A. List of symbols
A
A
B
C
"
E
D
F
H
H
I
K
L
¸
¸
M
P
Q
Q
Q
Q
Q
Q
S
S
"
area of open water (fraction)
minimum area of open water (fraction)
buoyancy #ux to the mixed layer, Wkg\
drag coe$cient (1.1;10\)
evaporation, m
thickness of layer by-passed by the convection, m
ice removed by freezing, m
depth of the mixed layer, m
depth of equilibrium mixed layer for which Q "(1!f )Q , m
volume of ice formed in open water, m
Newtonian transfer coe$cient, 1.43 J 3C\ m\
latent heat of melting, 335;10 J kg\
loss of latent heat to the atmosphere, W m\
latent heat of evaporation (2.50;10 J kg\)
ice melt, m
precipitation, m
oceanic loss of sensible heat, W m\
heat loss to the atmosphere, Wm\
loss of sensible heat to the atmosphere, W m\
heat loss to the atmosphere through the ice, W m\
heat loss at the open sea surface, W m\
heat entrained from the lower layer, W m\
salinity of the mixed layer
salinity of the lower layer
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
S
S
T
¹
"
¹
¹
¹
¹
;
b
c
c
d
d
f
f
f
g
k
m
q
q
r
t
t
t
t
u
H
v
v
w
w
w¹
wS
a
b
d
e
i
j
o "o
o
o
o
*o
salinity of sea ice
salinity #uctuations
temperature of the mixed layer, 3C
temperature of the lower layer, 3C
freezing temperature (!1.93C)
air temperature, 3C
temperature of ice surface, 3C
temperature #uctuations, 3C
wind speed, ms\
buoyancy, ms\
heat capacity of sea water ("3.98;10 J kg\ C\)
heat capacity of air ("1.0;10 J kg\ C\)
ice thickness, m
prescribed initial ice thickness ("0.1 m)
fraction of oceanic heat going to ice melt
fraction of oceanic heat leading to minimum ice melt rate
normalised fresh water content in the mixed layer
acceleration of gravity ("9.8 m s\)
thermal di!usivity of water ("1;10\ m s\)
dissipation factor ("1.25).
speci"c humidity of air
saturated speci"c humidity of air
f (f )\
time, s
time needed to reform the equilibrium mixed layer, s
time to "rst convection, d
time until the ice is melted, d
friction velocity, m s\
ice volume, m
maximum ice volume, m
entrainment velocity, m s\
vertical velocity #uctuations, m s\
turbulent heat transport
turbulent salt transport
coe$cient of heat expansion ("0.4;10\)
coe$cient of salt contraction ("8.0;10\)
freshwater content in the mixed layer, m
buoyancy parameter (e"1 when B'0, e"0.05 when B(0)
coe$cient of heat conduction through ice ("2.03 C\ m\ s\)
scale factor used during ice melt, m\
density of freshwater ("1000 kg m\)
density of water ("1028 kg m\)
density of air ("1.3 kg m\)
density of ice ("910 kg m\)
density step at the base of the mixed layer
1423
1424
B. Rudels et al. / Deep-Sea Research II 46 (1999) 1385}1425
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